Stellar Structure A star has a stable configuration.
That is, there is a certain structure (mass distribution) to allow
for such a force balance.
Inward = gravity Outward = gas pressure (gradient)
(ideal gas, degenerate gas) + magnetic pressure (mag = Τ2 8) +
radiation pressure (rad = Τ44 3) + turbulence pressure (tur = Τ2
2)
How is the pressure sustained? Energy thermal pressure
How is the energy generated?
How is the energy transported? 2
Hydrostatic equilibrium
Boundary conditions: → 0 and → 0 as → 0 → 0, → 0, and → 0 as →
∗3
Variables: , , , , , , , ,
Vogt-Russell “theorem” Given hydrostatic and thermal equilibrium
with energy produced by nuclear reactions, the internal structure
of a star, and its subsequent evolution, is uniquely determined by
the mass and chemical composition of the star.
In fact, … by any two variables above, cf. the HRD. It is not
really a “theorem” in the mathematical sense, i.e., not strictly
valid. It is a “rule of thumb”. There are other factors, too, such
as magnetic field or rotation, though these usually have little
effect.
4
= c
= 1
The velocity of the modulation, the group velocity =
;
Harwit 5
Hydrostatic equilibrium
()
2
= −
2 − 1
The LHS is usually null, unless there is free fall or
explosion.
6
Mass continuity
= 42 r … (2)
42
: total mass inside radius total = gas + e + rad
Using mass as the independent variable,
= , is preferred because mass is
usually given and fixed (but is not.)
Boundary conditions (1) at = 0, = 0, (2) at = or = , = 0.
7
8
= const
= 0, = const isobaric process
For an ideal gas
= 1 isothermal process
→ ∞ isochoric (= isovolumetric) process
2 , : degree of freedom
The specific heat capacity =
=
For an ideal gas, = 3, so = Τ5 3 ≈ 1.66
For a diatomic gas, = 5, so = Τ7 5 ≈ 1.40
For a photon gas, = 6, so = Τ4 3 ≈ 1.33 10
(1)
and Poisson eq.
or 2
(1) Gravity
Solution = −
(2) Electrostatics
Gauss’s law, = free, = , = −φ,2 = − Τ
Solution = −
4
Assume a polytrope; i. e. , a spherical fluid with and being
related by
≡ 1+ 1
is dimensionless and specifies how density varies with mass
1
2
2
1
2
= −
This is the Lane-Emden equation of index , after J. H. Lane and R.
Emden.
Compared to (3), a given a solution with different , and 0
a family of solutions
is dimensionless and specifies how radius varies with mass.
, Τc
= 1 and
= 0 at = 0, and can be integrated from = 0.
For = 0, 1, 5, analytic solutions are available; otherwise the
integration is done numerically.
= 0, 0 = 1 − Τ2 6
= 1, 1 = sin /
= 5, 5 = 1 + Τ2 3 − Τ1 2
For = 0 and = 1, solution → 0 at some point ( → 0); this defines
the boundary of the star, i.e., at first zero (1)=radius. Solve n 1
= 0.
For = 0, = c0 =const; for = 5, solution never goes to 0.
0 = 1 − Τ2 6 = 0 1 = 6
1 = sin / = 0 1 =
5 = 1 + Τ2 3 − Τ1 2 1 = ∞
15
= 0, a constant density sphere; 1 = 6; =
= 1, solution a sync function; 1 = ; = ; = 2
= 5, finite density, but infinite radius; 1 → ∞
16
Mass
=1
Radius = 1
=1
2
For = 5, → −∞. For any > 5 (i.e., < Τ6 5), > 0, the system
is not gravitationally bound; no stable configuration
Given a solution , i.e., , the density and pressure profiles can be
derived.
18
=1
−2
density
19
20
For the integration constants, 1 must be zero to avoid singularity
at origin.
Because = c at = 1, 2 =1
= 1 − 1
The case for = 0, = c0 = const.
1
2
= 0
1
22
If the star is supported by both radiation pressure and gas
pressure, the total pressure = gas + rad.
Define = Τgas .
=
Eliminate , = Τ , into 4 = Τ3 1 −
= Τ4 3 = Τ4 3 or = 3
This is the Eddington standard model ( = 3). 23
A special case --- an isothermal gas sphere ∝
This is a polytrope of = 1, or → ∞
>5, so the sphere is infinite in extent. Need to work out the
solution from beginning.
Recall Eq. 3, 1
= −4
Plug in the ideal gas equation of state, = Τ
1
2
2
4
= −
with the BCs, = 0, Τd d = 0 , and = 0.
This must be solved numerically, and the solution diverges (i.e.,
density never goes to 0, and mass goes to infinitive.)
Conclusion: A finite star cannot be an isothermal gas sphere.
25
28
Star Formation in a Nutshell
Stars are formed in groups out of dense molecular cloud cores.
Planets are formed in young circumstellar disks. (Jeans
criteria)
Initial gravitational contraction leads to a decrease of
luminosity, while surface temperature remains almost unchanged.
(Pre-main sequence Hayashi track)
Radiative
Convective
29
30
Zero-age main sequence (ZAMS): the locus in the HRD of stars of
different masses first reaching the main sequence (i.e., starting
steady core H fusion)
31
Novotny
in [K]
• Eddington in 1920s hypothesized that fusion reactions between
light elements were the energy source of stars.
• Stellar evolution = (con) sequences of nuclear reactions
• kinetic ≈ ≈ 8.62 × 10−8 ~ keV,
but Coulomb barrier = 122
[fm] ~ MeV.
This is 3 orders higher than the kinetic energy of the
particles.
• Tunneling effect in QM proposed by Gamow (1928, Z. Physik, 52,
510);
applied to energy source in stars by Atkinson & Houtermans
(1929, Z. Physik, 54, 656)
33
Resonance very sharp peak in the reaction rate
So there exists a narrow range of temperature in which the reaction
rate ↑↑ a power law
an “ignition” (threshold) temperature
For a thermonuclear reaction or a nucleosynthesis (fusion) process,
the reaction rate is expressed as
∝
Resonance reactions Energy of interacting particles ≈ Energy level
of compound nucleus
38
22 2
[cm−3s−1]
As , 12
Major reactions are those with smallest 12 , i.e., lowest Coulomb
barriers.
is the particle volume number density, = , where
is the mass fraction
39
0 1 0.0
1 1 0.0
1 amu = 931 Mev/c2
nucleus consisting of 1 p+ and 1 n0
Earth ocean 1.6× 10−4
Τ • 156 ppm … Terrestrial seawater 1.56 × 10−4
• 22~26 ppm … Jupiter • 17 ppm … Saturn • 55 ppm … Uranus • 200 ppm
… Halley’s Comet
45
+ → 4 + (destruction) faster
4
4 ≈
/2
/ ≈ 0.12 4 ≈ 2/9
The lower the mass density, the more the D abundant D as a
sensitive tracer of the density of the early Universe
46
7Li burns at ≈ 3 × 106 K
1H burns at ≈ 5 × 106 K
2 1, 3
7 1, 4
; =
This should be valid at the star’s center, thus
∗~ ∗
Stahler & Palla
Low-mass protostars, Tc too low to ignite Li fusion, so inherit the
full ISM Li supply.
Higher-mass protostars can burn and destroy Li promptly, but the
base of the convection zone is below 3 × 106 K, so the surface
lithium abundance = ISM value.
Stahler & Palla
Ca I λ6718 prominent in late-type stars
52
53
Stars Τ M > 0.08, core H fusion Spectral types O, B, A, F, G, K,
M
Brown Dwarfs
0.065 > Τ M > 0.013, core D fusion 0.080 > Τ M > 0.065,
core Li fusion Spectral types M6.5–9, L, T, Y
Electron degenerate core
< 3 × 106 K
53 1 ≈ 1000
Star
ZAMS
Birthline
The baryon number, lepton number, and charges should all be
conserved.
All 3 branches operate simultaneously.
pp I is responsible for > 90% of stellar luminosity
pp I important when
= 1.44 × 2 + 5.49 × 2 +12.85 = 26.7 MeV
= 26.7 – 0.26 × 2 = 26.2 MeV 6 × 1018 erg g−1
56
0.420 MeV to the positron and neutrino (0.26 MeV); position and
electron (each 0.511 MeV rest energy) annihilate 1.442 MeV
released
This neutrino carries away 7.2 MeV
This neutrino carries away 0.26 MeV
(5.493 MeV, 6 s)
(1.586 MeV)
(0.861 MeV)
(17.347 MeV)
(0.135 MeV)
(18.074 MeV)
Among all fusion processes, the p-p chain has the lower temperature
threshold, and the weakest temperature dependence.
= (4 − ) 2
(4 − ) : mass deficit
But some energy (up to a few MeV, depending on the reactions) is
carried away by neutrinos.
57
+ → 2 (unstable)→ +
Hans Bethe (1939) realized that the weak interaction was capable of
converting a proton to a neutron (!) first
Weak interaction a very small cross section
The neutron is more massive, so this requires energy, i.e., it is
an endothermic process, but neutron + proton deuteron (releasing
binding energy, i.e., is exothermic)
58
… but the nucleus of deuterium, a deuteron, consists of a proton
and a neutron!
The thermonuclear reaction rate is
= 3.09 × 10−37 2 6
Τ−2 3 exp −33.81 6 Τ−1 3
(1 + 0.0123 6 Τ1 3 + 0.0109 6
Τ2 3 + 0.0009 6) [cm−3s−1],
where the factor 3.09 × 10−37 2 = 11.05 × 1010 2
2
= 2.38 × 106 2 6
Τ−2 3 exp −33.81 6 Τ−1 3
(1 + 0.0123 6 Τ1 3 + 0.0109 6
Τ2 3 + 0.0009 6) [erg g−1s−1]
59
60
• PP I vs PP II
That is, 3He to react with 3He at a lower temperature,
or to react with 4He at > . ×
• Relative importance of each chain Branching ratio , ,
• Above > 3 × 107 K, PP III should dominate, but in reality, at
this temperature, other (CNO) reactions take over.
• The overall rate of energy generation is determined by the
slowest reaction, i.e., the first one, with reaction time 1010
yrs
~26.73 MeV (≈ 6.54 MeV per proton)
~ 1, ~ 4 − 6 ~ 6 for T ≈ 5 × 106 K ~ 3.8 for T ≈ 15 × 106 K (Sun) ~
3.5 for T ≈ 20 × 106 K
Recognized by Bethe and independently by von Weizsäcker
CN cycle + NO cycle
Cycle can start from any reaction as long as the involved isotope
is present.
CN cycle more significant
after that carried away by the neutrinos
61
CN cycle takes over the PP chains near T6=18. Helium burning starts
~108 K.
Schwarzschild
CNO dominates in stars > 1.2 M, i.e., of a spectral
type F7 or earlier large energy outflux a convective core
This separates the lower and upper MS.
62
bottleneck
16, 20, 24 … (the α-process)
He-burning ignites at Tc ~ 108 K
Nucleosynthesis during helium burning C12 , O16, = 7.162 MeV O12 ,
Ne16
63
C-burning ignites when Tc ~ (0.3-1.2) × 109 K, i.e., for stars
15-30 M
O-burning ignites when Tc ~ (1.5-2.6) × 109
K, i.e., for stars > 15-30 M
The p and α particles produced are captured immediately (because of
the low Coulomb barriers) by heavy elements isotopes O burning Si
64
= 2.4 × 106 2 6 −2/3 exp −33.8 6
−1/3 erg g−1 s−1
= 8 × 1027 6 −2/3 exp −152.3 6
−1/3 erg g−1 s−1
Clayton
−3 exp −42.9 8 erg g−1 s−1
≈ 4.4 × 10−8 2 3 8
40 erg g−1 s−1 (if 8 ≈ 1)
∝ 2 4
∝ 16
= 0.02 ok for Pop I
65
Photoionization
Photodisintegration
66
For example, 16O + ↔ 20Ne +
If < 109 K → but if ≥ 1.5 × 109 K in radiation field ←
So 28Si disintegrates at ≈ 3 × 109 K to lighter elements
(then recaptured …) until a nuclear statistical equilibrium is
reached
But the equilibrium is not exact a pileup of the iron group nuclei
(Fe, Co, Ni)
which can resist photodisintegration until 7 × 109 K
67
nucleon (MeV)
He 3α 100 C, O 0.61
C C + C 600 O, Ne, Na, Mg 0.54
O O + O 1,000 Mg, S, P, Si ~0.3
Si Nuc. Equil. 3,000 Co, Fe, Ni <0.18
From Prialnik Table 4.1
Interactions among charged particles Coulomb barrier
If there are enough neutrons around neutron capture, not limited by
Coulomb barrier, so proceed at relatively low s ever heavier
isotopes or radioactive decay
a new element +e− (beta decay) + (antineutrino)
Stable nuclei: neutron captures
Unstable nuclei: neutron capture or − decay
− decay has a constant time scales
0 capture time scales (, ), so may proceed slower (s-process) or
more rapidly (r-process) than the competing − decays
70
Nuclear reactions: mass to energy (light)
The reverse, energy into mass, is also possible; e.g., a photo an
electron + a position, if > 22, with the presence of a
nucleus
≈ ≈ 22, ≈ 1.2 × 1010 K
In reality, at 109 K, sufficient photons (tail of the Planck
function) for pair production. Annihilation immediately destroys
the positrons.
56 + 124 MeV → 13 4 + 4
If ↑↑↑, even 4 → + + 0
So stellar interior has to be between a few 6 and a few 9.
Lesson: Nuclear reactions that absorb (rather than emit) energy
from ambient radiation field (in stellar interior) can lead to
catastrophic consequences.
72
Luminosity
in electromagnetics, current density = conductivity [electric
field]
In hydraulics, [flow] [pressure gradient] / [resistance]
(unit) Pressure = [energy] / [volume]
∼ 42
≈
L (∝ 2eff 4 ) and eff
Main sequence is a run of and eff as a function of stellar mass,
with Tc nearly constant.
Why ≈ constant? Because onset of H burning ~107 K regardless of the
stellar mass
77
78
Recall for low-mass stars, ∝ 5.5 −0.5, pp chain ∝ c 4
The energy-generation equation,
3
4
= 6
7
~ Τ1 13 …………….. Stellar radius ↔ very weakly on the mass
~7 Τ1 13 ≈ 5.5… Stellar Luminosity ↔ strongly on the mass
79
Recall for low-mass stars, ∝ 5.5 −0.5, pp chain ∝ c 4
The energy-generation equation,
3
4
= 6
7
~ Τ1 13 …………… Stellar radius varies weakly with the mass
~7 Τ1 13 ≈ 5.5… Stellar Luminosity varies strongly …
In the HRD, ∝ 2 4 Τ981 1007 ∝
4
or log ≈ 4 log + const (i.e., constant radius)
80
For high-mass stars, ∝ 3, CNO cycle ∝ c 16
Then, 15 ∝ 19, so ∝ Τ76 9
or log ≈ 8.4 log + const
That is, a steeper MS slope in the HRD
log
log
Lower MS
Higher MS
Young (hot) population
L=30 kpc; H=200 pc H=1 kpc D=30 kpc
Bimodal population? Metal poorer
or
Calibrated with the Sun.
Energy can be transported by conduction or convection, or
radiation.
Conduction: by microscopic collision of particles and movement of
electrons. Flux density [erg/s/cm2] = −T
Convection: by bulk motion of particles in a fluid (gas or liquid):
advection (directional flow of
energy) or diffusion(non-directional along a
concentration gradient).
Stars transport energy by either radiation or convection.
Conduction is effective only in compact objects, e.g., in
isothermal cores in WDs.
NASA
86
1 ′, 1
′, 1 ′
Convection takes over? When an element moves vertically, does it
continue to move? Key: Temperature gradients
Element maintaining pressure equilibrium with surrounding, 2
′ = 2, ideal gas law → 22 = 2 ′ 2
′,
′ > 2 (or 2 ′ < 2) → sink back; no convection
To have convection, the element (rising adiabatically) should cool
slower than the surrounding (in radiative equilibrium), i.e.,
Radiation can no longer transport the energy efficiently
enough
Convective instability
The rising height is typified by the mixing length , or
parameterized as the scale height H, defined as the pressure (or
density) varies by a factor of e times. Usually 0.5 Τ
2.0 89
90
In meteorology, dry and cool air tends to be stable, whereas wet
and warm air (smaller gamma values) is vulnerable to convection
thunderstorm
Note that for an adiabatic process for an ideal gas
= ∝
Note , 93
How to calculate ad?
+ 1
for which = Τ5 3.
Note , ad
(convection to take place) is ( log
log ) < 0.4.
Convection occurs when rad > ad
That is, when rad is large, or when ad is small.
To recap
rad =
ad small large H2 dissociation (PMS Hayashi tracks) H ionization,
T~6,000 K He ionization, T~20,000 K He II ionization, T~50,000
K
96
1. Opacity ↑
2. e- receive energy d.o.f. , so g ad ⇒ susceptable to
convection
Development of hydrogen convective zones inside stars.
Similarly, there are 1st and 2nd helium convective zones.
97
For a sun-like star, ionization of H and He, and also
the large opacity of H ions a convective
envelope (outer 30% radius).
For a massive star 1.2 , the core produces
fierce amount of energy (via CNO) convective core
a large fraction of material to take part in the
thermonuclear reactions
For a very low-mass star 0.4 , ionization of
H and He leads to a fully convective star H
completely burns off.
98
A binary system at 5.74 pc. Gliese 752A (=Wolf 1055) is an M2.5 red
dwarf (mass ~0.46 solar, mV~9.13), whereas Gliese 752B (VB 10) is
an M8V (mass ~0.075 solar, mV~17.30).
99
T Tauri stars contracting down to the ZAMS an enlarged chromosphere
emission spectra
https://ase.tufts.edu/cosmos/view_picture.asp?id=174
102
∝ Τ∗ ∗ 3 ∝ ∗
−2
That is, lower-mass MS stars are denser at the cores to provide
sufficient pressure
So temperature may never get high enough for H fusion
Degeneracy important
Fermi-Dirac distribution for non-interacting, indistinguishable
particles obeying Pauli exclusion principle; applicable to
half-integer spin in TE. Examples of fermions include the electron,
proton, neutrons, and nuclei with odd mass numbers, e.g., 3He (2
e−, 2 p+, 1 n0)
Bose-Einstein distribution for particles not limited to single
occupancy of the same energy state. i.e., that do not obey Pauli
exclusion principle; with integer values of spin. Example bosons
include 4He, the Higgs boson, gauge boson, graviton, meson.
109
A Fermi gas is called degenerate if the temperature is low in
comparison with the Fermi temperature/energy.
110
• Temperature governs the flow of energy between two systems.
• Chemical potential governs the flow of particles; from higher
chemical potential to the lower.
111
112
113
In general, the pressure integral (momentum transfer)
= 1
3
114
∝ Τ5 3 (non-relativistic)
∝ Τ4 3 (extremely relativistic)
Particle in a Box
= Τ2 , = 1, 2, 3, …
Since = Τ = Τ
→ = Τ1 2 2 = 2 2 =
2
22
2 2 =
= 1 2
, 2 2
, 3 2
2
sin
At the center, 1, 3 probability max 2 probability = 0
c.f. classical physics: same probability everywhere in the
box
116
117
Fermi energy: the highest energy level filled at temperature
zero
= 2
Liquid 3He atoms 4 × 10−4eV 4.9
Metal electrons 2−10 eV 5 × 104
White dwarfs electrons 0.3 MeV 3 × 109
Nuclear matter nucleons 30 MeV 3 × 1011
Neutron stars neutrons 300 MeV 3 × 1012
Fermi energy of degenerate fermion gases
= 2
120
Non-relativistic
121
In the non-relativistic case
In the extremely relativistic case c in the pressure integral
122
In the NR case, ∝ Τ5 3~
3
log
3 log − 1.397
In the ER case, ∝ Τ4 3 = Τ4 3
4 , no solution between and .
A mass limit for a degenerate electron body (white dwarf)
Chandrasekhar limit Τ5.8
2
WD cooling: 11.47 ± 0.47 Gyr
Turn-off ~16 mag, 0.8 M
MS complete ~26 mag, 0.092 M
Detection limit ~30 mag, 0.083 M